Let $X$ be a locally compact Hausdorff space. A subspace $Y \subset X$ is locally compact (in the subspace topology) if and only if $Y$ is **locally closed** — that is, $Y$ is the intersection of an open set and a closed set in $X$.
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Equivalently, $Y$ is locally compact if and only if for every $y \in Y$, there exists an open set $V \subset X$ with $y \in V$ and $V \cap Y \subset Y$ with $\overline{V \cap Y}$ compact (closure taken in $X$).